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Shortest Paths in Portalgons

Authors Maarten Löffler, Tim Ophelders, Rodrigo I. Silveira, Frank Staals

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LIPIcs.SoCG.2023.48.pdf
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Author Details

Maarten Löffler
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
  • Department of Computer Science, Tulane University, New Orleans, LA, USA
Tim Ophelders
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
  • Department of Mathematics and Computer Science, TU Eindhoven, Tthe Netherlands
Rodrigo I. Silveira
  • Department de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Ophelders, Tim: Partially supported by the Dutch Research Council (NWO) under project no. VI.Veni.212.260.
  • Silveira, Rodrigo I.: Partially funded by MICINN through project PID2019-104129GB-I00/ MCIN/ AEI/ 10.13039/501100011033.

Acknowledgements

We are grateful to the anonymous reviewers for the detailed and valuable feedback provided, which helped us to improve the paper considerably.

Cite AsGet BibTex

Maarten Löffler, Tim Ophelders, Rodrigo I. Silveira, and Frank Staals. Shortest Paths in Portalgons. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.48

Abstract

Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric. We analyze the complexity of shortest paths. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons. The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Polyhedral surfaces
  • shortest paths
  • geodesic distance
  • Delaunay triangulation

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